Understand u substitution for integration (3 slightly trickier examples), calculus 1 tutorial
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- čas přidán 15. 06. 2024
- Calculus 1 tutorial on the integration by u-substitution, 3 slightly harder and trickier examples: integral of x/(1+x^4), integral of tan(x)*ln(cos(x)), integral of 1/(1+sqrt(x)).
Want more integral practice for your calculus 1 or calculus 2 class? Check out my "100 integrals" 👉 • 100 integrals (world r...
0:00 3 slightly harder and trickier integrals, calculus 1
Integral of x/(1+x^4), 0:47
Integral of tan(x)*ln(cos(x)), 4:29
Integral of 1/(1+sqrt(x)), 7:47
#calculus #blackpenredpen
I'm a first-year mechanical engineering major and let me just say you have SAVED my life.
the way i felt this 😭
How you doing?
@@damian4091 well thanks to this channel. 2.0 to a 3.8 gpa
@@unholykraut1107 jeeeeez good work
What
You can make u=cos x for problem 2 if you turn tan x into (sin x)/(cos x) because then you end up with the integral of (-1/u)(ln u) du and if you do another substitution where w=ln u you just get the integral of -w dw and after using the power rule and substituting u and x back in you will still end up with -1/2 (ln(cos x))^2 +C just takes a little longer.
yeah this was my first instinct as well
Dude you are a Genius, actually my favourite CZcamsr And my inspiration
Haha... nice black-red marker. I look at it the same way as I look at calculators... makes life much easier, but reduces the need for skill (mental arithmetic in the case of the calculator, and your previously demonstrated marker-switching skills, in the case of this dual colour marker.)
Great video!
Thank you thank you!
Hi, just a suggestion it's off topic but our Calc 2 course covers a section on using integration tables a d you're suppose to use substitution methods I think tonsolve. Were you ever going to do a hw set on those? It's the ones with complex integrals that can't be solved by hand.
Ty,
Dan
@@dmorgan0628 Ever figure those out? 😄
1:55 AM
1/21/2022
@@happyjohn1656 Nah I ended up taking a different course of life paths, ended up taking Calc 3, DIff eq, physics w/ calc and linear algebra and washed out hard that semester and said fuck it I'll go back to the work force. One day I'll relearn math for funzies and hopefully pass the classes I failed at.
@@dmorgan0628 can't wait for another life update after the next 4 years
Spoilers: he doens't keep using this. So it must not have worked out too well.
Too bad. He could have had all 3 colors in one hand easily if it did.
or he lost it
I saved it because it was a gift. I
@@blackpenredpen I kinda assumed you'd get more if it worked for you.
@Brooks Archer bro that is so toxic
best channel of math
Hey man, your videos are really good man. I couldn't understand u substitution for a long time, but you made me understand it within the first 10 minutes. Thank you!:)
Absolutely amazing video!!
I'm learning so much:)
And that marker is so perfect for you!
He only uses the marker in this video...
Thank you for this amazing challenge!
I have just been taught all 3 methods for integrals and im watching ALL your integral videos to learn. They are being extremely good, i now comprehend better how to do them! As always thank you and keep uploading videos, i love them! [And you too :)]
This was so helpful! Thank you so much!
br oyou´re by far the best maths teacher ever. i never got that substitution thing but now it makes perfectly sense. do you know a site with some training examples?
This is the best channel on the internet. Like seriously I fucking love this channel. So underappreciated!
Thanks. your work is helping many people.
Second example scared the sizzle out of me.
I wish if I had seen this before my math exam. Definitely coming back for more!
Very helpful, thank you!
Im first year pre-engineering student and i can't explain in words how grateful I am
thank you for the videos. They are very helpful
Love those integrals!
appreciate u man, u helped me pass my cal 1 final
Thank you man i appreciate your efforts
Vous m'avez appris beaucoup
Great video @blackredpen It does remember when I was studying systems engineering in a course called Mathematics II. It follow the success ando greetings from Venezuela
Thank you, Sherry. Here is what you were trying to read.
I used double substitution to solve 2 and 3 but I like how you do it with just 1 substitution
thank you sir, your hard work will never go unoticed...
Thank you!
Thanks Sheri! Very cool
this guy is the GOAT hands down
Yo did anyone else wake up at 9:31?
Great job! Love the new expo marker!
how beautiful!! Thank you :)
Very great video! My first day of calculus 3 was monday, and this was a great refresher for me on previous sections! Thanks! And if you can post some calculus 3 sections 11 and up, that would be greatly appreciated :) Nice marker btw haha
Hi there, sorry I am not teaching calc 3 anytime soon. (thank you for the comment regarding to the marker ^^ )
Thanks for explanation 😸🎉
Your Integration videos are addicting haha
Much better than the chalkboard and the two-headed marker sure looks easier to use.
So funny and still clever! :D
bprp and 3blue 1 brown are the only channels getting me thru calculus rn
this is an amazing video
I like the marker. Good marketing out there.
Please make a video on Euler's substitution and feynman's
Thank you :)
Thank you
Hey
Great video and awesome marker (haha). Anyways, I wanted to ask which book do you use for the questions?
Viraj Madaan i use Stewart for my classes. But oftentimes I just come up with my questions or search online.
Love you man!
شكرا لك الفيديو جدا مفيد 💞💐
In the question no. 2, we can write tanx as sinx/cosx and then put cosx = u and after substitution we have a very nice example of integration by parts! Haha
Put u = 1+x^2, du = 2xdx, xdx=du/2. u-world: Int(du/2u) = ln(u)/2 = ln(sqrt(1+x^2))+C I think.
Please don't forget to like the video. I watch all of these videos and they are so good, sometimes I forget to like them
How do you come up with that u-1 strategy ? How do you think like that ?
I'm convinced Expo only created that marker to get a marketing shout-out from this guy.
1:54 AM (yep!)
1/21/2022
😆
Hello, so for problem number 2, I got a different method but same answer. Set cosx equal to u, then change tanx to sinx over cosx, since they are the same. Derivative of cosx is -sinx, then it will be du/-sinx. This will cancel out the sinx (from sinx/cosx which is the same as tanx). Then it will become ln(u)/u, do substitution again, v equal ln(u), derivative is 1/u. Cancel out you at the bottom and so on.
The God of Math🙇🙇
😅 I was wondering we'll eliminate the √x
Genius
For question three, couldn't the integrand be rewritten as 1/(1 + (x^1/4)^2), and then you could just use the arctan integral?
Thank you, Sheri😅
For that 2nd question I will multiple -1 and -1 inside the integral then simple
For first part I took x^4 from denominator and after simplification I put 1/x=t but I am getting answer -1/4log|x^4+1/x^4| + c is it right.
Good!
the flash of thank your sharri made me spit my coffee out hahaha
4:27 Please tell me why don't people write acrtan(x)? tan^-1(x) confuses some of my friends cause they sometimes think that it's power.
would it not be better notation to write arctan instead of tan^-1?
Thanks for helping out
4:52 why don't division by sin cancell the tan
Good video
blackmarkerredmarker um can't you just put du right away and just replace the terms which match in the du?
Example: I = int((3+6x^2)(3x+2x^3)dx)
u = 3x+2x^3
du = 1*3x^0 + 3*2x^2
= 3 + 6x^2 dx
*Which means... *
I = int(udu) = (u^2)/2 + C
= (3x+2x^3)^2/2 + C
3rd part can also be solved by substituting x as (tan@)^2........I admit the fact that it requires further substitution of tan@ as t but the ans is same!!!
Amazing! \m/
In the last integral couldn't you factor out 2, and name C2/2=C3?
Hello thanks for your nice video, I have one question : at 13'48'' : 1 + square root of x is positive so there is no need to use absolute value...no problem if x > 0 but what happens if x < 0?
for x
and then u can divide everything by 2 because a constant divided by a constant is still a constant. Does not matter if you add constant numbers to each other or multiply/divide them
Try that and then take the derivative. You'll get a different answer than what you started with.
Let S equal the integral.
S=blah+c1
Divide everything by 2
(S/2)=(blah/2)+c2
You see that it's now a different answer, just like x doesn't equal x/2
simply because this is not an equation
7:47 i substituted u = sqrt x and i did the integral... i got direct answer i didnt need to merge that 2 into +c ....
When you solve an integral, are you allowed to merge all the constant into one single ”C”?
You're just adding a bunch of different constants, it'll still be a constant afterwards so it's fine
for oexams, make it clear what you are doing
Obviously.
@@chessandmathguy Get off your high horse buddy
C is just a random constant. So one C is enough. If you multiply anything with a constant you get the constant.
damn...thank u sir..
some smart guy
Actually nice sponsorship besides wonderful video!
I love your videos. My passion has always been for discrete maths but at the age of nearly 60 you've awoken a love of calculus.
In this one you state rt(x) is always positive. Surely not.
Elliott Manley rt(x) is always positive because the definition of a finding a root is basically just finding the number that has been squared. As you know, a number squared is positive e.g. -2^2 is +4
Therefore the reason for his statement is because you cannot square a number to get a negative number and nor can you do it in reverse (Unless you take into account complex numbers)
As an example rt(4) = both +2 and -2 for the reason explained making rt(-x) not true for all real integers for x
That's exactly my point.
Rt(4) = +/-2.
+Elliott Manley Rt(4) is not +-2. Rather, the solution to the equation x^2=4 is +-2. Rt(4) written on its own is strictly the positive result. I tell students: whenever you see a square root, it is positive. Whenever you have to take a square root to solve an equation, add plus or minus.
@@stephenbeck7222 , thanks. I was wondering about the same thing!
Could you please tell me why we have this convention? Is it to preserve a one-to-one mapping?
@@elliottmanley5182 sqrt(x) only finds the principal or positive root.
Kaique say: amazing video
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This 2 pointed marker is very 1 dimensional... What if we up it to 2 dimensions... a cross (think x-y axis)... with 4 marker tips, red, black, blue and green each at right angles to each other?... or 3 dimensions (think x-y-z axis), 6 points, red, black, blue, green, purple and yellow all at right angles?? ... Could we extend this into 4 dimensions? NOTE: If we consider a 3-d marker projected into a 2-d world... The 2-d world essentially "sees" the 3-d marker only 4 colors at a time.. when we rotate the 3-d, the 2-d world sees it as simply 2 of the 4 tips changing color... SO... on our 3-d marker set (6 points), if we could set the tips to change colors, we could essentially model a 4-d marker set where each 4-d orientation is "seen" in the 3-d world as the marker tips not only revolving but changing colors!... Could we model higher dimensions?... Would LOVE to see someone work out the details on this! ....Am I over thinking this black-red marker thing??
Why is it that square root of x is not invited in the u world? can you explain further
Tejero Life, square root of x is an expression in terms of x, we cannot integrate more than 1 variable in a single variable integral, the variable depends d(?)
Intégrale from 0 to π
Cos(nt)/1-sin(a)cos(t)
I personally believe that u-substitution is slightly trickier than IBP. Still, great video BPRP!
For problem 3 I did the u sub u = ×^1/2, so I ended up with the correct answer without the +2.
Since this was an indefinite integral that appears to not have mattered, but if this was a definite integral would that deeply affect the answer?
No because in definite integration, the constants will get subtracted and cancelled out (it's why we ignore +c when evaluating definite integrals)
i want the pen switching back!
I like it ,but what about using integration by parts on the second question
You can do the second one with integration by parts. It's a looper and regrouper in one.
Given: integral tan(x) ln(cos(x)) dx
Let tangent be integrated, and the log composition be differentiated.
d/dx ln(cos(x)) = 1/cos(x) * -sin(x) = -tan(x)
Construct IBP table:
S ____ D _________ I
+ ____ln(cos(x)) __ tan(x)
- ____-tan(x) _____ -ln(cos(x))
Attach S-column signs, construct along diagonal. Then construct an integral along the bottom row.
-ln(cos(x))^2 - integral tan(x) ln(cos(x)) dx
Spot the original integral, and call it I. Set the whole expression equal to I.
I = -ln(cos(x))^2 - I
Solve for I:
2*I = -ln(cos(x))^2
I = -1/2*ln(cos(x))^2
Solution:
-1/2*ln(cos(x))^2 + C
What is the integral of (e^x)(3^x)
Great video 10ks
my hero black red pen.
U r amazing BRoooooo
11:05 - Isnt integral of 1, x ? And why at 8:50 ,there is no more 1 over 2sqrtX ? But great vid , saving me before exams :D
The integral is taken in terms of u substituting in for x, so when taking the integral/antiderivative of 1 it would be u rather than x. Hope this helped!
A bit late for exams but the integral was in terms of "u" (notice the "du" at the end of the equation) so the antiderivative of 1 was u. And for at 8:50 he multiplied both sides by 2sqrt(x) so that du=(1/2sqrt(x))dx simplifies to 2sqrt(x)du=dx (again notice the difference in "du" and "dx")
For example 1, how will I know to make a u substitution of x^2 (especially in a exam), are there any methods/tips on knowing what to substitute ?
In this particular example the thought stream may go like this: "hmmm, we have to make x^4 something in terms of u, but there is this freaking x on the top. What can we do? We must get rid of it. Well, let's take something u-related, which after computing the derivative gives us something like x dx, then the x's will cancel out. How can we achieve x^1 term via derivation? Of course - by taking x^2." And then it goes.
The key to finding these ways out is to compute hundreds of integrals, and eventually they'll start popping out in your head automatically :)
A non-universal method for integrating with the chain rule is to say that int f(g(x))g’(x)dx= int f(g(x))d(g(x)). You don’t need to do any differentiation, and you can integrate x/(1+x^4) using this method.
great marker that you could habe made with 2 MARKERS AND A PIECE OF MASKING TAPE!!!! This guy works at an engineering school?
the way I would've done the second one is I would've done u = cos (x) because tan(x) is just sin(x)/cos(x) so you have du/u. Then you have integral of 1/u ln u which I'd recognise already, but you can also take it to the v world with v = ln u
Now of course this is essentially a more complicated way to do the same thing because v = ln u = ln (cos x) which is what you did, but I think it's a more understandable method to get there
Great video! After like 12 minutes it seems we have worn the ring from hobbit XD
Pls. Deep pen use
In 2 ques it will be 1/4 if u integrate udu
I want to buy one of ur tshirts. Can u help me to buy it? Sir
Output of square root of x not always positive. He can be zero and positive, so he always non-negative. But certainly square root of x+1 is always positive
By parts would work for the second one,